system of equation in terms of z

Question

1. x-3y+z=-2

2. -4x+9y=7

Answer 1

We can solve this system of equations by isolating one of the variables. We are asked to find the solution in terms of z, so it will probably be easier to solve for x or y first. 

 

Let's isolate x first. We don't want to worry about z yet, but lucky for us, -4x + 9y = 7 has no z, so we can start with this one. 

Part 1: We solve for x in the bottom equation:

 

-4x +9y = 7   --->  4x - 9y = -7  ---> 4x = 9y - 7  --->  x = 9/4 y  - 7/4.

 

[for the first arrow, we multiplied both sides by -1; for the second, we add 9y to both sides; for the third arrow, we divide by 4]

 

 

Part 2: Now, we replace x in the top equation with our solution for x in the line above:

 

x - 3y + z = -2  --->  (9/4 y - 7/4) - 3y +z = -2  ---> 9/4 y - 7/4 - 12/4 y + z = -8/4  ---> -3/4 y + z = -8/4 + 7/4

 

---> -3/4 y = -1/4 - z  --->  -3y = -1 -4z  --->  y = 1/3 + 4/3 z.

 

[for the first arrow, we substitute the equation that we found above for x; for the second, we multiply each coefficient by 4/4 to make adding the fractions easier; for the third arrow, we simplify and move 7/4 to the right hand side of the equation; for the forth arrow, we simplify further and move z;  for the fifth arrow, we multiply both sides by 4; for the sixth arrow we finish solving for y]

 

 

Part 3: Now, we can finally find x in terms of z. 

 

We plug the solution to Part 2 into the solution for Part 1:

 

x = 9/4 y - 7/4   --->  x = 9/4 ( 1/3 + 4/3 z ) - 7/4  = 3/4 + 3z - 7/4 = 3z -1.

 

 

So, our answer is: 

 

x = 3z -1,

 

y = 4/3 z + 1/3.